Pretty self explanatory really. Yarmulka or kippah is a minhag that we have to cover our heads. May Jewish actors remove their head covering for the purposes of their jobs?
If one can’t take a job unless he doesn’t wear a Kippah at work, one doesn’t have to forfeit his job for this mitzvah. If they allow one to wear a regular hat one must wear such a hat. Also, when one enters another room or the marketplace one must put back on a Kippah even if one will be mocked as long as there is no concern of losing one’s job
I think, in the above, even though it says "can't take a job", I infer that since the end of that sentence says, "doesn't have to forfeit his job", they mean that whether one is offered a job or is already working in that job, he don't need to wear the kippah if it damages his job.
I am inferring that in your question, the person's job is an actor, and the boss has specifically told him that he must play that specific role that requires not wearing a kippah. If he does not play that role, he will lose his job. If that's the case, then it would follow Halichipedia's position.
On the other hand, it seems that if they offer the person a choice of playing another role where he can wear his kippah, it seems that he should accept the other role. As stated above, it seems that the best solution might be to request if he can wear a hat, as non-Jews also wear hats / caps. That may keep the boss and him happy and satisfy halacha.